Preferred nanocrystalline configurations in ternary and multicomponent alloys

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Regular Article

Preferred nanocrystalline con

ﬁgurations in ternary and

multicomponent alloys

Wenting Xing, Arvind R. Kalidindi, Christopher A. Schuh

⁎

Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

a b s t r a c t

a r t i c l e i n f o

Article history: Received 18 August 2016

Received in revised form 7 September 2016 Accepted 11 September 2016

Available online 20 September 2016

In nanocrystalline alloys, a range of configurations can have low energies when solute atoms have favorable in-teractions with interfaces. Whereas binary nanostructured alloys have been well studied, here we lay ground-work for the computational thermodynamic exploration of alloy configurations in multicomponent nanocrystalline alloys. Multicomponent nanostructured systems are shown to occupy a vast space, with many to-pological possibilities not accessible in binary systems, and where the large majority of interesting configurations will be missed by a regular solution approximation. We explore one interesting ternary case in which thefirst alloying element stabilizes grain boundaries, and the second forms nano-sized precipitates.

Keywords: Ternary alloys

Grain boundary segregation Nanostructure stability Monte Carlo

Reducing grain size into the nanoscale regime is a promising path-way to improve the properties of engineering materials, such as achiev-ing a highﬁgure of merit in thermoelectric materials[1–3], improving the coercivity of magnets[4–6], and reaching superior mechanical prop-erties such as high strength and wear resistance[7–11]. Attaining and retaining grain sizes in the nanoscale regime, however, is contrary to the normal tendency for rapid grain growth when such a large volume fraction of grain boundaries is present. Stabilizing nanostructure is gen-erally most plausible in alloyed systems, where, e.g., an alloying element that exhibits a strong preference for grain boundary sites is introduced

[12–23]. In such an alloy system, grain growth is not just kinetically im-peded, but can be thermodynamically unfavorable if the dissolution of the segregant into the crystal lattice is an energy-raising proposition.

The role of the spatial distribution of alloying elements (i.e. the alloy configuration) on the stability of nanostructured systems has been widely studied in binary alloys, using a variety of approaches that in-clude ideal solution, regular solution, or other configurational assump-tions[24–30]. The lattice Monte Carlo approach of Chookajorn and Schuh[28]permitted a more stochastic approach to alloy configuration without enforcing ideality or regularity, and produced a similar set of predictions. This included the identification of four different classes of stable binary states in positive enthalpy of mixing systems: a grain boundary segregated state, a duplex state with both grain boundary segregation and solute precipitation, a state with no grain boundary segregation but with solute precipitation, and a bulk state with phase separation. The nanocrystalline states are not predictable from a bulk phase diagram, but result from the possibility that the solute atoms

can segregate to grain boundaries and attain a different energetic state than in any bulk phase.

In ternary alloys where interface states are allowed, the number of different accessible nanostructure configurations is expected to be sub-stantially larger, because the topological complexity available with three species is much higher, and the two solute elements can have ad-ditional interactions that lead to nontrivial configurations. Consider, as an example, an alloy characterized by simple nearest-neighbor pairwise interactions among elements A, B, and C. If the neighbor pairs are also subclassified as either crystalline (superscript ‘c’) or grain boundary (su-perscript‘gb’) bonds, then a binary system would have six different bond energies, EAAc , EBBc , EABc , EAAgb, EBBgband EABgb, while a ternary one would

require those plus an additional six, ECCc , EACc , EBCc , ECCgb, EACgb, EBCgb. Since the

relative magnitudes of these energies should determine the equilibrium con_{ﬁguration, the addition of more terms leads to an even more drastic} increase in the number of relative orderings of the energies they repre-sent. In general a total of n(n + 1) bonds in an n-ary nanocrystalline alloy system result in a total of [n(n + 1)]! different bond energy order-ings, and by extension unique conﬁgurational classes. Applying the con-straint that the like-atom grain boundary bonds should always have a higher energy than the like-atom crystalline bonds, we estimate that a maximum of½nðnþ1Þ!

2n different alloy conﬁguration classes exist in

nanostructured n-ary alloys.

The black circular data points inFig. 1show the consequence of this scaling for n-ary alloys; we immediately see that the ternary problem in nanocrystalline alloys is many orders of magnitude more complicated than the binary problem, in the number of nominally distinct system conﬁgurations that are achievable. On the other hand, we may also ex-pect some signiﬁcant degeneracy among these states; in any given

⁎ Corresponding author.

E-mail address:schuh@mit.edu(C.A. Schuh).

http://dx.doi.org/10.1016/j.scriptamat.2016.09.014

Contents lists available atScienceDirect

Scripta Materialia

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ordering of the bond energies we might imagine that only the lowest ones are relevant to the equilibrium structure, and higher energies will correspond only to activated states that are rarely, if ever, sampled, and only when entropy or geometric constraints from other elements are important. For example, the four clearly distinct conﬁgurations re-ported by Chookajorn and Schuh in the binary alloy model were dictat-ed by only the two lowest bond energies for each element[28], as compared to the nominal ~200 predicted inFig. 1. If we limit our discus-sion to the number of orderings of the r lowest-energy bonds for each element in the system (r denoting the“rank” of the bond orders consid-ered), the number of states is considerably reduced. This is also shown inFig. 1, where we count the number of subsets of size r of the bond en-ergy set, and all permutations within those subsets under the constraint that Egb_NEc_{for like-atom bonds of all elements.}

An algorithm that enumerates the number of distinct bond order-ings is provided in the Supplementary material, and the results of such calculations are shown for different ranks inFig. 1. As expected, fo-cusing attention on the most energetically favorable terms reduces the number of possible distinct nanostructured alloy con_ﬁgurations dra-matically. However, the main point remains: the ternary problem is more than an order of magnitude more complicated than the binary one in any case, and adding components to the problem raises the num-ber of possible conﬁgurations exponentially. Of the 2016 different com-binations in the 2nd rank ternary model, 1854 could be nanocrystalline states (i.e. at least one of the two lowest-energy bonds of an element is a grain boundary bond), and thus a large space of interesting nanostruc-tures may exist as thermodynamic equilibrium states that have yet to be explored.

On the other hand, we can consider the number of distinct nanostructural configurations that may be achieved under the regular solution assumption, i.e., forced random mixing. The regular solution as-sumption only allows for grain boundary segregation and solid solution states, and does not permit ordering; it therefore can address only very few configurations compared to an unconstrained system. So far as we are aware, the only works considering multinary nanostructured sys-tems have relied on this simplifying assumption[27]or an even stricter assumption of ideal solution behavior[23], but in addition to being physically questionable, we see inFig. 1that the regular solution simpli-fication misses much of the interesting complexity in multinary

systems. Hundreds or thousands of distinct conﬁgurations are possible when mixing is permitted to be nonrandom, and we expect that this is the most interesting space for investigation of ternary nanostructured alloys.

The results inFig. 1motivate greater study of ternary and multinary nanostructured alloys, and imply a rich array of alloy con_{figurations that} could be of scientific and technological value. However, they also show that the space of possibilities is vast and complicated, so thermodynam-ic methodologies that can both rapidly scan the space while capturing the most essential physical features and avoiding the most limiting as-sumptions are needed to address this problem. We propose that lattice Monte Carlo simulations of the kindfirst proposed by Chookajorn and Schuh[28]and applied to a number of binary systems[31–35]may tribute to the rapid assessment of the ternary and multinary alloy con-figuration space. Rather than address the entire space in the present letter, we take a_{first step to illustrate the opportunities for alloy design} in nanostructured ternary alloys. We perform a case study where the majority solute (B) prefers a bulk, phase separated microstructure and anti-segregates from grain boundaries in the solvent (A), while the mi-nority solute (C) has a strong preference for grain boundary segregation and prefers a nanocrystalline state. Accordingly, the bond energy order chosen is: EgbABNE gb BCNE gb CCNE gb AA=BBNEcAC=BCNEcCCNE c ABNE c AA=BBNEgbAC ð1Þ

The bond energies can be related to actual alloy systems according to the enthalpy of mixing and enthalpy of grain boundary segregation for each element pair, as per the method in Ref.[28], where increasing the enthalpy of grain boundary segregation of an element pair generally leads to a decrease in the corresponding grain boundary pairwise bond energy and increasing the enthalpy of mixing leads to an increase in the corresponding crystalline pairwise bond energy. Here, the bond energies (shown inTable 1) are chosen to reﬂect enthalpies of mixing of ΔHmix

AB ¼ ΔH mix AC ¼ ΔH

mix

BC ¼ 20 kJ=mol and enthalpies of

grain boundary segregation ofΔHseg

AB¼ −45 kJ=mol, ΔH seg

AC¼ 65 kJ=mol,

andΔHseg

BC ¼ 10 kJ=mol.

We use the Monte Carlo method of Chookajorn and Schuh[28]to identify the equilibrium alloy configuration of this system. In this ap-proach, each lattice site has a chemical identity– either an A, B, or C atom_{– and a grain number such that adjacent lattice sites with different} grain numbers have a grain boundary pairwise bond energy. Two Monte Carlo events are used to explore the configuration space of this alloy: an atom swap where two different types of atoms are randomly swapped, and a grain swap where the lattice sites at the grain boundary can change their grain number to that of an adjacent grain or take on an entirely new grain number. For this study, compound forming alloys are not considered in order to avoid the need for compound units and multi-body interactions[36]. Starting from a very high temperature of 10,000 K and slowly lowering it to 773 K through 100,000 Monte Carlo steps, the simulation equilibrates at the lowest free energy alloy configuration considering both nanocrystalline and single crystalline alloy configurations on a 200 × 200 × 6 BCC lattice. Although the simu-lations are three dimensional, we show two dimensional cut sections throughout the paper. The Python code for this simulation is provided

Fig. 1. The number of distinct alloy conﬁguration classes based on the number of alloying elements in the system. This is calculated considering all pairwise bond energies (full rank), and also considering only the lowest‘r’ bond energies (rank ‘r’).

Table 1

Bond energies for corresponding simulations. Corresponding

ﬁgures

Bond energy [meV/bond]

EAA/BBc ECCc EABc EAC/BCc EAA/BBgb ECCgb EABgb EACgb EBCgb

Figs. 2 and 3(c) 0 31.2 25.9 41.5 50 81.2 218.4 −51 91.5 Fig. 3(a) 0 31.2 25.9 41.5 50 81.2 218.4 62.9 91.5 Fig. 3(b) 0 31.2 −25.9 41.5 50 81.2 114.7 −51 91.5 Fig. 3(d) 0 31.2 25.9 41.5 50 81.2 44.8 −51 91.5

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in the Supplementary material and for more details the reader is re-ferred to the original work of Chookajorn and Schuh[28].

Fig. 2a shows the solvent-rich corner of the ternary phase diagram under consideration, along with a series of nanostructures equilibrated at 500 °C at a variety of compositions inFig. 2b. The A–B binary edge of this system has a bulk, phase separated ground state, and at 500 °C ex-hibits solubility of B in A to a saturation limit around 5 at.%. Conversely, the A-C binary edge evolves a grain boundary segregated nanocrystal-line state, wherein smaller average grain sizes are stabilized with in-creasing amounts of solute C. This behavior is denoted by Chookajorn and Schuh[28]as a“classical nanostructure” and is the most well-stud-ied case of a grain boundary segregation-stabilized nanostructure in the literature.

In the ternary alloy, the binary A–B and A–C states appear to roughly superimpose; the equilibrated states that emerge inFig. 2b all exhibit grain boundaries decorated with C and B-rich precipitates. In fact, the equilibrium grain size is apparently prescribed by the concentration of C, declining in a manner similar to that seen in the A–C binary system and essentially independent of the amount of B. However, the C-enforced nanocrystalline state disrupts the tendency for B to aggregate into a single precipitate as favored by the bulk thermodynamics of the A–B binary. Instead, element B forms precipitates within nano-grains deﬁned by C-rich boundaries. Due to the grain boundary anti-segregating tendency of B, a core-shell structure within the grains forms with the region of A atoms separating the precipitate of B atoms from the grain boundary which is saturated with C atoms (shown more clearly inFig. 3c).

The energetic preference for a nanostructured state in this ternary alloy is conﬁrmed by comparing the internal energy of this structure with that of an equilibrated structure in a single crystal as shown in

Fig. 2(c–d). The single crystal reference state is assessed at the same composition by equilibrating a structure in which interfacial states are not allowed and a single grain number is enforced over the whole struc-ture. Overall, it is clear that the system favors the formation of a nano-structured state due to the substantial energetic beneﬁt for C atoms to reside at grain boundary sites. It is further noted that all elements re-ceive an energetic beneﬁt from forming the nanostructure in this case:

since C atoms are no longer dissolved in the A matrix, A atoms see an en-ergetic relief; and B atoms, though forced to form a number of nano-pre-cipitates instead of one coarsened precipitate, form a larger total volume of B precipitate, which lowers the internal energy of B. The increase in precipitation is attributed to the unique core shell grain structure, where 15% of the lattice sites are occupied by grain boundary atoms. As a result, B atoms are only able to occupy 85% of the system, which in-creases the effective supersaturation of B, leading to more precipitation. In addition, although solute C prefers to segregate to the grain bound-aries and solute B prefers anti-segregation, we do notﬁnd that the pres-ence of B negates the energy reduction of the grain boundaries of A due to the presence of C, as has been previously expected analytically[6]

based on a regular solution assumption. This is due to the independence of B and C and their ability to locally order in non-regular arrangements that are energy lowering.

In terms of the bond energy framework, the lowest two bond ener-gies for each element largely dictate the formation of the duplex grain structure inFig. 2(b). Speciﬁcally, C atoms at grain boundaries are pre-ferred by EACgbbEAAc (for A) and EACgbbECCc (for C) whereas second phase B

precipitation is favored by EBBc bEABc (for B). Different possible 2nd rank

orders will, in most cases, produce different equilibrium conﬁgurations; for instance,

a. when the two lowest bond energies for A and C are crystalline with a tendency to phase separate– EAAc bEACc and ECCc bEACc – a

nanocrystal-line state is no longer the minimum free energy conﬁguration and bulk thermodynamics prevails.

b. when the lowest bond energies for B are reversed to be EABc bEBBc ,

sol-ute precipitates of B no longer form.

These two cases are respectively shown explicitly inFig. 3(a) and (b), whereΔHAC

seg

is lowered to 21 kJ/mol for theﬁrst case and ΔHABmixis lowered to -20 kJ/mol for the second, with all other

ther-modynamic parameters maintained from the simulations inFig. 2

(corresponding bond energies shown inTable 1).

Fig. 2. (a) A schematic of the solvent-rich portion of the ternary composition space with (b) corresponding equilibrated nanostructures (B atoms in black, C atoms in white, and colors denote different grain allegiance). (c) The internal energy of equilibrated nanostructured states (2 at.% C atoms) are compared to equilibrated single crystal states at different concentrations of B; (d) the internal energies are broken down to each element for the 20 at.% B and 2 at.% C alloy.

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However, considering all 2nd rank orders in this alloy is not necessarily suf_{ﬁcient either, and one such case is shown in}Fig. 3(c–d). The microstructure inFig. 3(c) is the same as the one in

Fig. 2(xB= 0.2 , xC= 0.02), with a core-shell duplex grain structure.

Alternatively, the enthalpy of grain boundary segregation of B could be positive, which changes the bond energy order in Eq.1

from EABgbNEAA/BBgb to EABgbbEAA/BBgb :

Egb_BCNEgb CCNE gb AA=BBNE gb ABNE c AC=BCNEcCCNEcABNEcAA=BBNEgbAC ð2Þ

This produces a microstructure without the_{“core-shell” grain} struc-ture, as shown inFig. 3(d) (ΔHABmixis increased to 22 kJ/mol), where the

precipitates of B are now in direct contact with grain boundaries and in addition now interact with C. The lowest two bond energies for each element are the same in both cases, and it is not until the 4th lowest bond energy of B is considered that the difference between these two classes of microstructure is explained: the bond energy orders for B be-come EBBc bEABc bEBCc bEABgb(Fig. 3(b)) instead of EBBc bEABc bEBCc bEBBgb. Such a

high bond energy becomes relevant in this situation because the presence of grain boundaries is imposed by solute C; thus the grain boundary bond energies (i.e. whether B segregates or anti-segregates) become relevant, whereas in a binary nanocrystalline alloy such a case cannot exist since the single solute element must stabilize the grain boundaries.

In conclusion, we have evaluated the relative complexity and ﬂexi-bility afforded in nanostructure design by traversing from simple binary to multicomponent systems. In ternary and higher order alloys, many unique microstructures are expected to become thermodynamically plausible, with complex topologies that are not amenable to simple mean-ﬁeld or regular solution modeling. Combinations of elements with complementary roles in shaping the microstructure should open the door to new design strategies as well. Our Monte Carlo simulations herein show a few interesting examples from among the many thou-sand that are nominally possible, including core-shell grain boundary segregated states and dual-phase nano-duplex structures in which the length scale is set by the ternary addition. In light of the recent rising

interest in impurity effects in nanostructured alloys[23,33,34,37,38], a simple non-regular multinary model such as the one presented here could provide signiﬁcant insight on necessarily complex experiments in this space. It is hoped that future work can further develop the range of viable multinary nanostructures and seek them in the laboratory.

Acknowledgements

The authors are grateful for theﬁnancial support by U.S. Army Re-search Of_{ﬁce under grant W911NF-14-1-0539 and through the Institute} for Soldier Nanotechnologies at MIT under contract number W911NF-13-D-0001. A.R.K. acknowledges a National Defense Science and Engi-neering Graduate Fellowship.

Appendix A. Supplementary data

Supplementary data to this article can be found online athttp://dx. doi.org/10.1016/j.scriptamat.2016.09.014.

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